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11    THE SPECTRAL THEORY OF HUBERT        411

(b)    With the notation of Problem 15, let C be the operator of rank 1 defined by
C - x=:-(x | e_i)e0 • Show that Sp(t/4- C) is the disk |£| ^ 1 while Sp(C7) is the circle
|£| = 1. (Consider separately the restrictions of U + C to the subspace generated by
the en with n ^ 0 and to its orthogonal supplement.)

(c)    Let N be a normal operator on E and C a compact operator on E. If Sp(AO is
nondenumerable, show that the same is true of Sp((W 4- C)*(N -f C)) (use (a) above and
(15.11.8(i)). Deduce that the one-sided shift operator V (Problem 19(a)) cannot be of
the form N+C (observe that K*F= 1B).

21.    Let E be an infinite-dimensional separable Hilbert space.

(a)    Show that every nonzero two-sided ideal $ of the ring & (E) contains the ideal
<5 of operators of finite rank. (If T ¥= 0 belongs to $, show that every operator of rank
1 can be written in the form BTC for suitably chosen operators B and C.)

(b)    Show that the only closed two-sided ideal of the Banach algebra -S?(E), other
than -$?(E) and {0}, is the ideal & of compact operators. (First observe that <£ is the
closure of @, and then that if a two-sided ideal contains a noncompact operator, then
it also contains a noncompact positive hermitian operator H (Problem 6). For such
an operator, show that there exists an interval M = [a, H- oo [ with a > 0 such that
the space E(M) (in the notation of (15.11.9)) is infinite-dimensional. If Vis an isometry
of E onto E(M), show that V*HVi& invertible.)

22.    Let E be an infinite-dimensional separable Hilbert space. An operator with index on E
is a continuous operator T such that (1) T(E) is closed and of finite codimension;
(2) r-1(0) is finite-dimensional.

(a)    If T is an index operator, show that there exists a continuous operator A such
that 1E — A Tand 1E — TA are of finite rank. (Show that Tis a homeomorphism of the
orthogonal supplement F of J^O) onto T(B)9 and take A to be the inverse homeo-
morphism on F and zero on the orthogonal supplement of F.)

(b)    Conversely, suppose that Tis a continuous operator on E, for which there exists
a continuous operator A such that 1E — AT and 1E — TA are compact. Show that T
is an index operator, (Using (11.3.2), show first that the kernels of T and T* are finite-
dimensional, and hence that T(E) has finite codimension. Then use the fact that the
restriction of AT to the orthogonal supplement F of the kernel of AT is a homeo-
morphism onto its image, and finally use (12.13.2(iii)).)

23.    Let E be an infinite-dimensional separable Hilbert space.

(a)    Let T be a continuous operator on E such that T~l(Q) is of infinite dimension.
Then E is the Hilbert sum of an infinite sequence (En)n>0 of infinite-dimensional sub-
spaces, such that the Ert with n S> 1 are contained in 71" *(()). For each n ^ 1, let Sn
denote an isometry of E0 onto En. Let A be the continuous operator which on E0
is equal to Si, and on En is equal toSn + iSnl9 for all n ^ 1. Also let Fbe the operator
which is zero on E0, is equal to Srl on EI, and is equal to Sn-j.Snl on £„ for all
n i> 2. Let TQ denote the restriction ofPEQTto E0, and let B be the continuous oper-
ator which is equal to VT on E0, to — ToSj;1 on EI, and to —Sn,iT0Snl onEnfor
all n i> 2. Prove that T = AB — BA.

(b)    Deduce from (a) that for any continuous operator T7 on E there exist four
operators A, B, C, D such that

T = (AB - BA) + (CD - DC).

(Write T as the sum of two continuous operators, each of which has an infinite-
dimensional kernel.)s that -1 is an eigenvalue of T~1C.)uch that (jitw) converges vaguely to a measure which does not belong to M£(X, v).